Orville composed functions f(x)=4x−−√3−2 and g(x)=(x+24)3 to see if the two functions are inverses of each other. His work is shown below.f∘g=f(g(x))f∘g=f((x+24)3)f∘g=4(x+24)3−−−−−−√3−2f∘g=4(x+24)−2f∘g=(x+2)−2f∘g=xWhich of the following is true about Orville’s work?

Orville is checking whether the two functions \( f(x) = \sqrt[3]{4x - 2} \) and \( g(x) = (x + 24)^3 \) are inverses of each other. To verify this, we would typically compute both \( f(g(x)) \) and \( g(f(x)) \) and see if they both return \( x \), indicating that the functions undo each other.

Let’s examine his steps:

### Step 1: Checking \( f \circ g \) (i.e., \( f(g(x)) \))

1. **Starting with \( g(x) = (x + 24)^3 \):**
   He correctly substitutes \( g(x) \) into \( f(x) \) for the composition:
   \[
   f(g(x)) = f((x + 24)^3)
   \]
   
2. **Substitute into \( f(x) \):**
   Since \( f(x) = \sqrt[3]{4x - 2} \), we substitute \( (x + 24)^3 \) into this formula:
   \[
   f((x + 24)^3) = \sqrt[3]{4((x + 24)^3) - 2}
   \]
   
   However, Orville makes an error here by simplifying incorrectly. The correct expression should remain as:
   \[
   f(g(x)) = \sqrt[3]{4(x + 24)^3 - 2}
   \]
   
   This cannot be simplified further without additional steps, but Orville simplifies it inappropriately in his next steps, leading to an incorrect conclusion.

### Conclusion:
Orville's simplification contains errors. He should not have simplified \( f(g(x)) \) as \( x \), since the functions are not inverses in the form they are given. Therefore, the correct answer is:

**Orville made an error in simplifying \( f(g(x)) \). The two functions are not inverses of each other.**

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Do you want more details or have any questions?

Here are five questions you might explore next:
1. How do you properly check if two functions are inverses of each other?
2. What are the algebraic steps needed to simplify \( f(g(x)) \) correctly?
3. Can you find \( g(f(x)) \) and verify if it returns \( x \)?
4. What properties must a function have to be invertible?
5. How can you graphically verify whether two functions are inverses?

**Tip:** When simplifying compositions of functions, make sure to apply operations step by step without assuming results early in the process.

Orville composed functions f(x)=4x−−√3−2 and g(x)=(x+24)3 to see if the two functions are inverses of each other. His work is shown below. f∘g=f(g(x)) f∘g=f((x+24)3) f∘g=4(x+24)3−−−−−−√3−2 f∘g=4(x+24)−2 f∘g=(x+2)−2 f∘g=x Which of the following is true about Orville’s work?

Explore how to verify if two functions are inverses by composing f(x) = √[3]{4x - 2} and g(x) = (x + 24)^3. This problem involves algebraic simplification and function composition, ideal for students in Grades 10-12.

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